Last updated at May 29, 2018 by Teachoo

Transcript

Prove 1 + 2 + 3 + . + n = (n(n+1))/2 for n, n is a natural number Step 1: Let P(n) : (the given statement)\ Let P(n): 1 + 2 + 3 + . + n = (n(n+1))/2 Step 2: Prove for n = 1 For n = 1, L.H.S = 1 R.H.S = (n(n+1))/2 = (1(1+1))/2 = (1 (2))/2 = 1 L.H.S = R.H.S P(n) is true for n = 1 Step 3: Assume P(k) to be true and then prove P(k+1) is true Let P(k): 1 + 2 + 3 + . + k = (k(k+1))/2 be true We will prove P(k+1) is true, 1 + 2 + 3 + . + (k+ 1) = ((k+1)( (k+1)+1))/2 1 + 2 + 3 + .+ k + (k+ 1) = ((k+1)(k+2))/2 We will prove (2) from (1) From (1) 1 + 2 + 3 + . + k = (k(k+1))/2 Adding k + 1 both sides 1 + 2 + 3 + .+ k + (k+ 1) = (k(k+1))/2 + (k+1) 1 + 2 + 3 + .+ k + (k+ 1) = (k(k+1)+2( +1))/2 1 + 2 + 3 + .+ k + (k+ 1) = ((k+1)(k+2))/2 Which is the same as P(k + 1) P(k+1) is true when P(k) is true Step 4: Write the following line By the principle of mathematical induction, P(n) is true for n, where n is a natural number

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.